Why some people say it's true: That is the first derivative test we were taught in high school.

Why some people say it's false: There are cases that are exceptions to this statement. Also, I was taught that this is just the first step of the first derivative test.

The statement is \( \color{red}{\textbf{false}}\).

Knowing that a function's derivative is 0 at a specific point does not necessarily mean that there's a local minimum or maximum at that point. When \(f'(x) = 0,\) it only implies that there is a horizontal tangent at that point. An example of a case when a function has a horizontal tangent at a point that is not a local extremum is given below.

Additionally, it's possible for a function to have a local minimum or maximum at a point where its derivative is non-zero in a variety of different ways outlined further below. For example, \(f(x) = |x|\) has a local minimum at \(x = 0,\) despite \(f'(0) \neq 0.\)

Counterexample 1:

This is a simple counterexample of the given statement: an example of when the derivative of a function is 0 at a point which is not a local extremum.

Consider the example of \(f(x) =x^3\), which is continuous and differentiable everywhere (for all \(x \in \mathbb{R}\)).

Differentiating \(f(x)\) with respect to \(x,\) we have \(f'(x) = 3x^2 .\) Setting it to zero, we get \(x = 0.\) But graphically we can clearly see that the function does not attain a local extremum at \(x = 0.\) The curve merely goes from being concave downwards for \(x < 0\) to being concave upwards for \(x > 0.\)

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Counterexample 2:

\(y = \sin x\) for \( 0 ≤ x ≤ \dfrac{\pi}{3}\) is an example of a function which has a local extremum at a point where its derivative is not 0. This occurs because the function has a restricted domain.

Rebuttal: The gradient of the function is positive at \(x = \dfrac{\pi}{3}\), and therefore it is increasing
and the value we got isn't the local maximum. Reply: The function isn't defined for \(x > \dfrac{\pi}{3} \) , and hence we have indeed reached the local maximum at \( x = \dfrac{\pi}{3} .\)

Rebuttal: Isn't \(-1\) the minimum value of \(\sin x?\) Reply: The minimum value of the function \(\sin x\) in the domain \( 0 ≤ x ≤ \dfrac{\pi}{3} \) is not \(-1.\) The value \(-1\) corresponds to the minimum value on a larger domain.

True or False:

The \(\textbf{local extrema}\) of \(f(x)\) are the points where \(f'(x)=0 \).

False
True

The graph above depicts the function \(\color{darkred}{f(x)} = |\cos x + 0.5|\) in the interval \(\color{darkred}0 \leq \color{darkred}x \leq \color{darkred}{10} \). How many local extrema does the function \(f(x) \) have if its domain is restricted to \(\color{darkred}0 \leq \color{darkred}x \leq \color{darkred}{10} \)?